Is the word esse the same in Brazilian and European Portuguese?

No

The difference between the two varieties lies in usage and grammar. In Continental Portuguese, there is a strict spatial distinction between este (near the speaker), esse (near the listener), and aquele (far from both). In Brazilian Portuguese, this distinction is much more fluid. Brazilians frequently use esse/essa/isso to refer to objects or ideas that are physically close to the speaker, where a Portuguese person would strictly use este/esta/isto.

Brazilian Portuguese Examples

1. Esse celular é muito caro. (This cell phone is very expensive.)
2. Você viu esse menino ali? (Did you see that boy there?)
3. Eu não entendi esse assunto. (I didn't understand this subject.)
4. Esse doce está uma delícia. (This sweet is delicious.)
5. Me passa esse controle, por favor. (Pass me that remote, please.)

Continental Portuguese Examples

1. Este telemóvel é muito caro. (This mobile phone is very expensive.)
2. Viste esse miúdo ali? (Did you see that boy there?)
3. Não percebi este assunto. (I didn't understand this subject.)
4. Este doce está uma delícia. (This sweet is delicious.)
5. Passa-me esse comando, por favor. (Pass me that remote, please.)